New decay estimates for linear damped wave equations and its application to nonlinear problem

<jats:title>Abstract</jats:title><jats:p>We present new decay estimates of solutions for the mixed problem of the equation <jats:italic>v</jats:italic><jats:sub><jats:italic>tt</jats:italic></jats:sub>−<jats:italic>v</jats:italic><jats:sub><jats:italic>xx</jats:italic></jats:sub>+<jats:italic>v</jats:italic><jats:sub><jats:italic>t</jats:italic></jats:sub>=0, which has the weighted initial data [<jats:italic>v</jats:italic><jats:sub>0</jats:sub>,<jats:italic>v</jats:italic><jats:sub>1</jats:sub>]∈(<jats:italic>H</jats:italic><jats:sup>1</jats:sup><jats:sub>0</jats:sub>(0,∞) ∩<jats:italic>L</jats:italic><jats:sup>1,<jats:italic>γ</jats:italic></jats:sup>(0,∞)) × (L<jats:sup>2</jats:sup>(0,∞)∩<jats:italic>L</jats:italic><jats:sup>1,<jats:italic>γ</jats:italic></jats:sup>(0,∞)) (for definition of <jats:italic>L</jats:italic><jats:sup>1,<jats:italic>γ</jats:italic></jats:sup>(0,∞), see below) satisfying γ∈[0,1]. Similar decay estimates are also derived to the Cauchy problem in ℝ<jats:sup><jats:italic>N</jats:italic></jats:sup> for <jats:italic>u</jats:italic><jats:sub><jats:italic>tt</jats:italic></jats:sub>−Δ<jats:italic>u</jats:italic>+<jats:italic>u</jats:italic><jats:sub><jats:italic>t</jats:italic></jats:sub>=0 with the weighted initial data. Finally, these decay estimates can be applied to the one dimensional critical exponent problem for a semilinear damped wave equation on the half line. Copyright © 2004 John Wiley & Sons, Ltd.</jats:p>